Toroidal Vortex Filament Knots & Links: Existence, Stability and Dynamics

TL;DR

This paper constructs and analyzes the stability of vortex knots and links on a torus using a specific model, providing explicit formulas and numerical validation, with implications for stable vortex configurations.

Contribution

It offers a full characterization of the stability of toroidal vortex knots and links, including explicit formulas and stability boundaries, advancing understanding of vortex filament dynamics.

Findings

Toroidal vortex knots can be explicitly characterized for stability.

A ring of more than 7 filaments can be stabilized with a slight twist.

Filament lattice configurations are stable for all rotation frequencies.

Abstract

Using the Klein-Majda-Damodaran model of nearly-parallel vortex filaments, we construct vortex knots and links on a torus involving periodic boundary conditions and analyze their stability. For a special class of vortex knots -- toroidal knots -- we give a full characterization of both their energetic and dynamical stability. In addition to providing explicit expressions for the relevant waveforms, we derive explicit formulas for their stability boundaries. These include simple links and different realizations of a trefoil knot. It is shown that a ring of more than 7 filaments can potentially be stablized by giving it a slight twist and connecting neighbouring filaments on a torus. In addition to rings, (helical) filament lattice configurations are also considered and are found to be dynamically stable for all rotation frequencies and also energetically stable for sufficiently fast rotations. Numerical simulations are used to compare the Klein-Majda-Damodaran model with the full three-dimensional (3D) Gross-Pitaevskii equations as well as to confirm the analytical theory. Potential differences between the quasi-one-dimensional and the fully 3D description are also discussed.